Updates on the measurement of the speed of gravity using LDVs and MEMS resonators

Five years ago, at the XIII AIVELA conference, a project was presented by the author for the laboratory determination of the speed of propagation of the gravitational interaction using a vibrating tungsten disc as source of a local gravitational perturbation and a high-Q silicon resonator as gravitational antenna. Using laser Doppler vibrometers to track the vibrations of the transmitter and of the receivers, the speed of propagation of gravity would have been calculated from their measured phase difference. Numerous developments happened in the project since then, from the construction and acquirement of important components of the generator of dynamic Newtonian fields, to the experimental measurement of the speed of VHF radio waves in the near-field region as a preparatory technical test for handling and elaboration of the corresponding gravitational data. This latter experiment produced unexpected superluminal results, making it worthy of further study. The results of the test also broadened the scope of expectations and possible interpretations of the results of the gravitational experiment, including the additional role of an indirect assessment of the existence of gravitons.


Introduction
At the XIII AIVELA conference a proposal was presented for the measurement of the speed of propagation of the gravitational interaction using Laser Doppler Vibrometry on two gold-plated silicon resonators driven gravitationally to mechanical resonance by the dynamic Newtonian field generated by a massive disc of tungsten vibrating at the first modal frequency of the MEMS resonators [1].
Work on the experiment was later suspended during the long pandemic disruption and because of a personal scientific detour of the author, anyway it was recently resumed, and the aim of this paper is to document the new developments, the most important of which are unexpected experimental results from a test designed to verify the data handling and elaboration procedures.The test was a measurement of the propagation speed of radio waves in the near field region of a VHF antenna.The expected result was a subluminar but relativistic speed, as predicted by Maxwell and Einstein.The measured speeds, exceeding the speed of light beyond the measurement error, were initially considered the result of an error in the measurement of the delay, but after checking for every possible error source, from software to hardware to the environment, the focus was changed on the characterization of the phenomenon, trying to maximize the discovered effect, getting regular, predictable superluminal signals.Verified signal propagation speeds up to 6 c were recorded over a 150 cm separation between the receiving antennas.A considerable amount of time was then spent trying to give the recorded phenomena a theoretical interpretation, finding five possibly related hypothetical interpretations.Before all of that anyway, in the year following the previous publication, several design changes were applied to the setup to be used in the gravitational experiment, and some components were custom built, others acquired.Newer, higher precision calculations were performed on the expected results, and errors were corrected.These calculations made it clear that advanced cryogenic refrigeration up to 0.1 K would have been needed for obtaining a barely acceptable signal to noise ratio of 10 dB.Existing cryogenic systems were analyzed and considered, and finally Leiden cryogenics was contacted because of their experience with gravitational experiments.This led to technical discussions about the specific needs of the experiment, followed by a meeting in Pisa between their founder, Giorgio Frossati, and the author, at the NEST Labs where Leiden cryogenics technology is applied for leading edge nanotechnology research.The problem of vibrations at the sample site in the dilution refrigerator was discussed at length and it was calculated that at the band of interest, 2.2 kHz, it would be sufficiently managed by existing technology.The requirements are still extreme anyway, as the amplitude of the gravitationally generated mechanical resonance in the antenna at 0.1 K would be around 10 fm.
Another crucial requirement for the experiment is the bandwidth of the generator.Because of the expected Q value of 5 • 10 6 for the antenna at 0.1 K, the frequency of vibration of the generator must match that of the antenna with a possible error of only ±22  as the antenna has a bandwidth of only 44 .
Considerable work has been done on the generator design and realization.A disc diameter of 35 cm was found sufficient if associated with a thickness of 1 cm, giving it a mass of 18.3 Kg.The disc was custom machined by Baoji Hanz Metal Material Co., Ltd. and delivered in late 2019.
The disc is made in tungsten with purity 99.95% and smoothly polished surfaces.Two 1:5 scale tungsten discs were also made for testing the suspension and actuation designs.The design of the latter systems proved to be very complex.
A small piezo transducer and a larger 40W disc-shaped actuator were acquired to test the actuation part of the control system with both the model discs and the actual one.That said, the design of the suspension system proved to be extremely difficult compared to the problem of actuation.
In the design presented in the previous paper, the support system was mechanical.Tungsten rods were to surround the disc in twelve radial directions, connecting the disc to the internal wall of the generator's vacuum chamber like metal springs holding a trampoline oscillator.In early 2019 the length of the tungsten disc was calculated so that the frequency of the generator would be that of the antenna, and 300 tungsten rods with diameter 0.15 cm and length 1.99 cm as well as 300 others with diameter 0.03 cm and length 3.23 cm were manufactured along with the discs, for both the main disc and the models.Fine tuning of the frequency was intended to be operated with a precise control of the temperature of the rods, or more generally of the temperature inside the vacuum chamber of the generator.The rods would have been laser welded manually and a special 3D printed holder for facilitating the alignment of the rods was later engineered by the author.
Several practical implications descended from this clever design for the support system (which was suggested to the author by Christian Rembe during the XIII AIVELA conference).The first was the relative simplicity of the control system as the required vibration would have also been the first modal of the mechanical system.A low electrical power could then be converted very efficiently into mechanical motion, and the electromagnetic noise emitted from the system would have been minimal and completely screened from the wall of the vacuum chamber.Anyway, there were two drawbacks that appeared only later but could not be ignored: the system had no tolerance for even the smallest error in the design and construction phases, as laser welding would permanently modify the tungsten disc and the vacuum chamber.Assuming such spotless construction could actually be achieved, the temperature of the whole vacuum chamber should have been kept to within ±1  of the temperature experimentally determined to cause the exact resonance frequency to meet that of the antenna.This is difficult to achieve as electromechanical parts would also be present in the vacuum chamber, including the actuator.
More flexible designs for the suspension and actuation of the generator have then been considered.The most promising so far seems to be an electrostatic levitation system.It should be installed inside the vacuum chamber to control both the position and the attitude of the disc and cause it to vibrate at the exact required frequency and amplitude.An advanced control loop can be created using two Attocube IDS 3010 laser diasplacement sensors, obtaining picometer precision over 6 axes, 4 of which would be perpendicular to the surface of the disc but offset from its centre, to measure its tilt.
The vibration of the central point of the disc, the crucial data for measuring the phase difference from the receiving antenna, should instead be measured using a Polytec laser Doppler vibrometer.
The clock of the controller of the Polytec LDV should be phase-locked with a reference clock.The same reference clock should also be connected to a second Polytec LDV that will record the vibrations of the receiving antenna.
Since both the receiver and the transmitter are within vacuum chambers, there are two possible solutions for monitoring their movement with LDVs.The first solution is keeping the LDV on a separate vibration-isolated table and directing the laser using optical windows made of sapphire.Some 0.1 K cryogenic chambers come already equipped with optical windows, like the tabletop dilution refrigerator XL PRO manufactured by Qino.Unfortunately, being a new product destined to quantum computing research, its vibration spectrum at the frequency of interest is not known yet.
The second solution is using cryogenic vacuum-compatible optical fibres and optical heads to connect the LDVs to the generator disc and to the receiving antenna.Using optical fibres anyway is known to reduce the performance of LDVs and could result in an insufficient spatial or temporal resolution of the measurement.It could also affect accuracy and the SNR level.
There is no perfect solution.Any real word experiment must be done with what is available unless a totally new instrument is designed and built, or an existing instrument is deeply customized by the manufacturer.
The goal of the measurement must be realistic, knowing that even a result with a 10% of uncertainty would bring a valuable insight into the nature of gravitational interaction.The results of the test with radio waves made it clear that unexpected results do not always come in the decimals but sometimes in the order of magnitude!

Test with radio waves
It has been known for a long time that gravitational experiments can be first tested in the electromagnetic domain, as the forces are much stronger and the problem of noise much more manageable.The classical electrostatic and gravitational fields are mathematically similar and even General Relativity can be reduced, for the experimental physicists working in a laboratory, to a set Maxwell-like equations, as pointed out by Robert Forward in 1961 [2].
Electromagnetically generated forces have always been used for technical tests and calibration in gravitational experiments.Continuing this tradition, a measurement of the speed of radio waves has been designed and performed by the author to test data handling and elaboration of the results of the gravitational measurement.The results anyway were more than surprising and resulted in a shifted focus of the presentation at the last AIVELA congress.

Description of the apparatus
A signal generator was connected to a monopole radio antenna, acting as transmitting antenna.Two other monopole antennas, acting as receiving antennas, were connected to channel 1 and channel 3 of the oscilloscope.The antennas were aligned and positioned at precise locations, and their distance was frequently checked.The receiver 1 connected to channel 1 was closer to the transmitter then the receiver 2. The distance between the transmitter and the receiver 1 was 0.956 m, while the distance between the receiver 1 and the receiver 2 was 1.499 m, equal to the wavelength of a 200 MHz radio wave.This length was chosen because it can be traveled by light in 5.0 ns, so it makes comparing the recorded delay to the speed of light extremely simple: any delay smaller than 5 nanoseconds means the signal was superluminal.Of course, when sampling at 4 GSa/s, a delay of 4.75±0.25 ns cannot be considered superluminal as 0.25 ns is the absolute error of the instrument.
While the general schematics of the experiment never changed, the instrumentation used for the radio test was substituted and improved during the various sessions of data taking, lasted about 3 months.The changes were necessary to check if the unusual result was due to defective instrumentation, and to increase the quality of the data improving the SNR and temporal resolution.

The generator.
The first generator used for the signal to be sent to the transmitting antenna was Liquid Instruments Moku:Lab.This is clever FPGA-based instrument that can also operate as a Lockin Amplifier with 120 dB of dynamic reserve as well as a Spectrum Analyzer, a Frequency Response Analyzer, and a Laser Lock Box, among others.It can generate sinusoidal waveforms with 200 MHz of maximum frequency and 13 dBm of maximum power.
The second signal generator used was the Rohde & Schwarz SMG 801.0001.52,an older unit but third-party tested before the experiment and still perfectly working.It has a frequency range up to 1 GHz and a power up to 13 dBm within specs or 16 dBm out of specs.

The antennas.
The antennas have always been monopole antennas with a BNC connection, but they were replaced multiple times.In the initial phase of the experiment Matedepreso 136-174 MHz rubber ducky antennas were used, while in the following phases, including a "parallel test" explained later, Bingfu dual band 25-460 MHz, 470-1200 MHz whip antennas with a magnetic base were used.
In all setups the same type of antennas was used for the transmission and reception of the signals.Monopole antennas are omnidirectional in the ground plane and their gain is usually < 2.1  anyway the actual gain of the antennas directly measured.In the most recent measurements, a receiving antenna which was less efficient than the others were substituted with a new but identical one.

The clock.
For many measurements a 10MHz GNSS-disciplined oven-controlled SC-cut crystal oscillator with an Allan deviation <5•10^(-12) at 1s has been used as a frequency standard to generate accurate sinusoidal waves to be transmitted by the antenna.The Rohde & Schwarz signal generator was sufficiently accurate by itself, as well as the oscilloscope.They both were tested anyway, by the author and by third parties.The oscilloscope was also brand new, and the settling time of the internal clock was respected.

The oscilloscope.
After a technical test with the Moku:Lab used also as oscilloscope, and the first day of recording with a 1 GSa/s sampling rate oscilloscope, it was immediately clear that an higher sampling rate was necessary because the signal was faster than expected.So, all further measurements were done with a new Rigol MSO5204 mixed signals oscilloscope with all upgrades enabled (350 MHz band, 200 Mpts memory depth).Only two channels were used, connected to the two receiving antennas.The sampling rate of many measurements was 4 GSa/s, but the latter measurements were made using different configurations allowing de-facto 50 GSa/s on two channels and the measurement of phase differences as low as 2 ps between the channels.This capability was crucial in the "parallel test".

Data recording and transfer.
Single waveforms were recorded on the oscilloscope's memory, and, if they passed a preliminary verification of the noise level and of the general correctness of the sampling, they were saved on a Samsung MUF-128BE3 pen drive.The single waveform displayed was saved as well as the full signal recorded in the memory of the instrument and the screenshot of the waveform to identify the set.Files were numbered sequentially and named in a descriptive way: session, frequency and power of the generator, special settings of the generator, special settings of the oscilloscope.Some recording sessions used a different method, calculating in the instrument the average values of up to 1000 sampled waveforms.
2.1.6.Data storage and elaboration.Data were stored and elaborated in a ThinkPad P52 laptop with a Xeon® E-2176M processor, 64 GB of ECC ram, 4 TB of NVMe storage, running Windows Pro for Workstations version 10.Data were elaborated in both Python and Matlab to increase the reliability of the results.Maple 2022 was also used, but more in the design phases.The most important results were verified with several different algorithms.Parameters calculated in Python and MATLAB from raw data were always confronted with those calculated internally by the oscilloscope.No significant difference was found, confirming the internal elaboration of the instrument can be used with confidence for this measurement.

Measurement procedure 2.2.1.
Measuring the noise spectrum.The first step was the measurement of the spectrum of background electromagnetic noise to find the cleanest bands.The results were reassuring distant signals were relatively weak in the VHF band so most frequencies from 150 MHz to 200 MHz could be used without fear of external interference.

Measuring the power drop of noise signals.
The second step was the sampling of waveforms from external signals in the frequencies of interest to check their power difference between antennas.Such power difference was found to be moderate and explainable on the different positions of the antenna with respect to walls and some internal differences in gain.

Measuring the local signals power drop.
The first measurements with locally emitted signals confirmed the drop of power was much more significant between the receivers as expected in the near field region of any antenna.

2.2.4.
Measuring the delay of noise signal.Noise signals were propagating luminally between the two receivers, sometime a larger delay than expected was recorded, but slower than light radio waves are not unexpected.The propagation speed in the far field is conditioned by atmospheric and meteorological conditions, like pressure, temperature, humidity, and pollution, which cause predictable variations in the refractive index of air.The urban environment adds a further level of complexity with possible reflections from the walls and the ground, and possible interference between reflected signals can happen too.

Measuring the delay of local signal
This is the measurement were things got interesting.It was not a single recording session but many long ones over 3 months, with progressive improvements in hardware, refinements in the settings and in the procedures.Both luminal and superluminal signals, up to 6.6 c have been measured.Once the author found out that superluminal results were not due to measurement errors, that they were not unheard of in scientific literature, and that they could have theoretical explanations, the focus of the test was shifted to the study this phenomenon in every possible detail, like the dependence on power or on frequency.The connection of the gravitational experiment was not lost anyway, it instead opened new realms.A superluminal gravitational propagation could be, according to this author, the first indirect proof of the quantization of gravity and of existence of gravitons, as several interesting explanations of the phenomenon come from quantum theories.

The parallel test
A significant suggestion came from Alexander Dräbenstedt at the XV AIVELA conference.To test if the delay was in some way related to antenna malfunctioning, the two antennas should have been tested at the same distance from the source.Because of their bases, they could not be put closer than 3.6 cm, but the second antenna was affectively placed parallel to the first.The setup left an impossible to remove difference of 0.7 mm from the source, equivalent to 2 ps at the speed of light.This delay was measured in 5 different waveforms during the parallel test, along with other extremely small delays, which confirmed that the antennas were functioning correctly.The parallel test gave other interesting results in a few recorded waveforms that deserve further study.As an example, one waveform had a negative delay of 20 ps, while others had a delay similar to the time it took a signal to reflect from one antenna to the other.An example of possible double reflexion was also observed.Of course, the parallel setup was not designed to detect superluminal signals, so it did not add data on that aspect.

Theoretical considerations
After the necessary premise that the author is not a theorist and that the following hypothetical explanations of superluminal signals were completely unfamiliar to the author until the experimental results appeared on his screen, it is noteworthy that there are at least 5 possible explanations to the observed phenomenon in the literature.
In addition to the cited hypotheses, the possibility that the results could have been caused by some unidentified source of error cannot be overlooked.It also necessary to mention the possibility that the superluminal results could have been caused by some unknown physical effect, phenomenon or particle.

Conclusions
The laboratory measurement of the speed of gravity gradually progressed from being an inspiring intellectual pursuit of a few theoreticians, like Bruno Ferretti [27] [28], to a challenging but feasible goal for experimentalists.
The analogue experiment carried out with radio waves opened an equally interesting path of research that could be as fruitful as the gravitational experiment, especially since the theoretical implications of the results of both measurements could potentially be revolutionary.